Hexagonal Blog

Good news everyone. The time has come, I believe, to announce the existence of this new and improved HEXNET site. The site has actually been up for several weeks now, and constitutes the third major iteration of Hexnet.org under the neohexagonal regime that began in 2010, replacing last year's tragically doomed Drupal 7 version.

First and foremost, as promised, I have finally moved OFF DRUPAL. Drupal is dead to me now, and shall forever remain so. Let us never discuss it again. Beyond that, the most obvious change, I think, is that the site now looks completely different. The main point I want to make about this is that the new layout is much closer than its predecessor to what I intended when I first started working on the Drupal 7 version last year. The ultimate result of that earlier redesign should be seen in retrospect as having been a tragic aberration. I became fixated with some very flawed ideas vis-a-vis how the UI should work, and pursued them long after it should have been clear that they weren't viable. Even after some much-needed fixes earlier this year, it remained, for me at least, practically unusable. It was a bad layout, I was wrong to make it, and I apologize for having done so. The new layout, conversely, should provide a pleasant and fully responsive experience on a wide variety of clients. I have tested it to my satisfaction on both iOS and Android browsers, and it is even marginally functional on IE 8 (for fuck's sake please stop using IE 8). There are definitely aspects of the design I consider sort of unfinished, and which I will continue working on in the coming weeks and months, but it meets my baseline requirement of presenting the site's content in a readable, navigable format that—hopefully—does not actively annoy the reader, and for this reason alone represents a vast improvement over its predecessor.

I want to take a moment to consider the conspicuous return recently of the so-called hexagon "trend." We first wrote about this phenomenon in a fairly forgettable post from 2010, but in the past year or so it seems to have resurfaced, more forcefully and more explicitly than ever, and we would be remiss in our obligations to the hexagonal community if we did not at least make note of it.

I don't really feel like carrying on with any sort of extended commentary on the matter this time—the trend speaks for itself, and its merits should be obvious to even the most casual adherent to hexagonal principles. It will suffice, therefore, to simply compile a list of recent articles on the subject, and let the reader draw what conclusions they will from them. Though it should be understood of course that we don't see this as a "trend" so much as a central—indeed, perhaps the central—turning point in the entire enterprise of human civilization. Our proverbial children and grandchildren, looking back on this era, will not be asking about the time hexagons were "trending," they will be asking about the time hexagons, as a concept, were finally rising above the clutter and minutiae of the collective human memescape to turn the very course of human civilization, putting us on our inevitable path towards a glorious and unimaginable hexagonal destiny.

We seem to have fallen a bit behind in terms of keeping this site updated with RELEVANT MATHEMATICAL INFORMATION about hexagons. This is a deficit we're looking to correct as soon as possible. First up, let us consider the permutohedra. We at HEXNET.ORG have been meaning to write something about permutohedra for a couple of years now, but have never really found a good opportunity to so. WE WILL NOT EXPEND GREAT EFFORT DOING SO NOW. It will suffice to merely describe the concept in conjunction with some helpful imagery, which will hopefully serve as a useful foundation for further geometrical observations and investigations in the near future.

Yes, I have returned from an AWKWARDLY LONG BREAK after my last blog post to once again announce HEXAGONAL AWARENESS MONTH. Since I do not, for the time being, want to be associated with this site at all, on any level, I have created a NEW SITE dedicated to this, the most glorious of all awareness months, mostly cobbled together from pieces of other sites I am currently developing. [Edit 2013-11: This is in reference to the previous, Drupal 7 version of this site, let us never speak of it again. The present incarnation is in fact mostly derived from the HAM site, which itself is no longer maintained and will be merged back in with the Hexnet site at some point. [Edit 2014-02: The HAM site has been slightly rehabilitated in anticipation of this year's festivities, and there are no immediate plans to eliminate it. Though it is still pretty lame.]]

It is fairly minimalistic, both like, in the good way, and in arguably the bad way. I am aware that the "forum" in particular, though keeping with the overall minimalism of the site, does not really come close to the baseline standards people expect from web fora in this day and age. Nonetheless it seemed preferable to confusing people with the "post index" terminology I have used here. I remain committed to my vision of content sites where there is no underlying difference between blog-style posts and forum-style posts, and look forward to more effectively implementing this scheme in the future. [Edit 2013-11: I no longer adhere to this paradigm at all, and have no intention of implementing it now or ever.]

Today would be a good day I think to take a break from not-updating this site to remind everyone that this year's Hexagon Project at Interdependence Day Scranton is currently underway. I hadn't been planning on writing another post about it this year, but of late I received this lovely promotional poster, which I thought at least worth relaying to the wider hexagonal community. In commemoration of this year's festivities, I have also added to the Hexagonal Library a PDF of a presentation by Hexagon Day organizer Beth Burkhauser to the National Art Education Conference, "Global Interdependence and Art Education: Where Hexagons Make the Connection." More musings on the project can also be found in last year's blog post on the subject. In general of course HEXNET.ORG supports all efforts to relate the civic virtues of hexagons to their more familiar geometric ones, as both are clearly manifestations of the same unified underlying reality.

We at HEXNET.ORG have been experiencing a rash of weird, bot-like comments on various and sundry posts over the past several days. We're not sure what caused this evil to be unleashed on us, but it's possible Hexagonbot is at least partly responsible, as it seems to have attracted a lot of bot attention on Twitter. Anyway, our previously airtight CAPTCHA has failed us, so until we resolve this, anonymous comments will be moderated. We hope to return to normal as soon as possible, but we simply do not have time to deal with this crap at present.

After literally hours of research and development at our new GLOBAL HEXAGONAL HEADQUARTERS in Worcester, Massachusetts, Hexnet is pleased to announce the release of the world's first fully-automated hexagonal awareness retweeting service: Hexagonbot v.1.0.

Hexagonbot is a companion service to our Hexagonal Awareness Project reblogging blog on Tumblr, and is part of a broader effort by the global hexagonal community to bring hexagonal services to a variety of blogging and social media platforms.

Hexagonbot implements a proprietary algorithm through which it downloads a feed of hexagon-related tweets, and then retweets them. It's a very sophisticated process. Hexagonbot is not currently programmed to interact with other Twitter users in any way, though we anticipate this functionality may be added to future iterations of the platform.

I first became aware of the Hexagonismo y Arquitectura blog earlier this week, when I saw it on a list of traffic sources to this site. At first I didn't know what to make of it, being a bit too Iberian for my linguistic competencies, but upon further inspection it seemed to be affiliated with a "Hexagonismo" movement operating out of Ibiza, more of whose work, I think, can be found at Hexperience.org. [Edit 2013-11: This site unfortunately seems defunct at this point.]

This 2001 article from the Colonial Williamsburg Journal offers a fairly eloquent exposition of certain aspects of hexagonal symbolism that are worth noting. I found the hexagons themselves fairly interesting too—particularly in contrast with the rather hexagon-poor architecture of colonial New England that I'm more familiar with. I've never been a great fan of Anglicanism, or High Church aesthetics in general, but the Puritans clearly couldn't design an interesting building if their lives depended on it—indeed, they probably would've interpreted a hexagonal cupola as a sign of idol worship or something.

We have launched a new reblogging blog on Tumblr for the purpose of raising hexagonal awareness and hexagon-consciousness among the Tumblr community. Going forward, I expect this blog to be an important aesthetic precursor to the eventual promulgation of HEXAGONALISM as a self-sufficient artistic tendency.

"Ten is doubtless a convenient number of fingers to have, though men have gotten along with less and a few people have been born with more. But as the purely arbitrary unit which determines the form of our numbers, it was a miserable choice." – F. Emerson Andrews

"Do not disturb my circles." – Archimedes

It's that time of year again—time to start thinking about DOZENAL PI DAY.

Overview

Regardless of its radical prejudices, Pi Day has become, in recent years, an annual cultural event of some magnitude, and the occasion for a certain nominal enthusiasm about "math" that people seem to find meaningful. At the very least, it seems to remind us of our own cleverness, which is something I have nothing against in principle.

Yet I am more convinced than ever that, unless we remain firmly mindful of what separates the arbitrary and symbolic in mathematics from the truly meaningful, Pi Day will ultimately remain a rather vacuous and misguided holiday. Thus, I once again propose the observation of DOZENAL PI DAY on 18; March (decimal 20th), to raise awareness of the superior dozenal radix, while also bringing attention to the underlying meaninglessness of our present decimal notation.

I feel something must be said about this astrological "sign shift" business that has set the 'tubes abuzz in recent days. There being twelve astrological signs ("Ophiuchus" notwithstanding) arranged around a central point, the issue is clearly one of both dozenal and hexagonal import. Despite the fact that I'm not all that into astrology (though I've learned enough over the years to inform my general fluency in Western occult symbolism), I feel the hexagonal principles at stake must be defended.

First of all, this is of course old news. I don't really understand why it keeps popping up in the media every few years, but this is simply the latest incarnation of a story that we've all been well aware of since classical antiquity, and that certainly should not come as news to anyone, anywhere. "Ophiuchus" has always been there, the twelve signs have never been symmetrically arranged in exact 30-degree sections, axial precession has always been going on, we always knew about it, nobody has ever cared, and nobody cares now.

In summary, for those who do not follow such things (and I have noticed that "graphene" is among the top keywords bringing people to this site, so it is quite possible you do follow such things): Graphene was "discovered," or if you will isolated, by Geim and Novoselov in 2004, by peeling off layers of graphite with scotch tape. It is essentially an indefinitely large aromatic molecule, and the flat, two-dimensional form of the buckyball or the buckytube. For a variety of reasons I won't get into here, it has numerous potential applications in electronics and nanotechnology, and is quite interesting all around.

It had never actually occurred to me prior to 2004 that graphene needed to be "discovered." I had always been taught that graphite, as a major allotrope of carbon, consisted of one-atom-thick sheets of carbon atoms arranged in a hexagonal lattice, and that this had been known for quite a while (since at least the advent of X-ray crystallography). Thus it really shouldn't have been too much of a conceptual leap to assume that, in fact, such sheets existed (though apparently the prevailing view was that the sheets would "roll up" when isolated). I still have a hard time wrapping my mind around the fact that it took until 2004 to actually discover this. Particularly with all the research that's gone on with fullerenes since the '80s, you'd think somebody, somewhere, would've taken the time to actually isolate a sheet of graphite. Apparently it is somewhat difficult, but crap I have scotch tape and pencils lying around, by all rights that Nobel could've been mine.

I have recently been made aware of a contest being held on Kotaku Australia regarding "hexagons that should not be." Apparently inspired by the hex tiling in Civilization V, they are giving away a free copy of the same to whoever most impressively photoshops a hexagonal shape onto an object that is not normally hexagonal. Unfortunately, the contest ends Monday night, so there's very little time to spread word of this to the larger hexagonal community. I wish I'd heard about it sooner. Anyway, samples of submitted work can be found on the following pages, and are most interesting:

[Edit: The winners and runners up can be found here and here, respectively.]

I find it fascinating, obviously, both as a Civ fan (still trying to get 5 running under Wine!) and as a hexagonalist. Creative exercises of this sort, inducing people to think artistically about the roles and potential roles of hexagons in everyday life, are potentially very valuable tools in expanding hexagonal consciousness. It reminds me in some ways of last month's Interdependence Day Hexagon Project—in both cases, the hexagon is explicitly employed not for some incidental aesthetic reasons, but for its very hexagonality itself. This can only be, I think, a positive trend in our society, as we as a species come to terms with the implications of our glorious hexagonal future. I am hoping to see more and more of this sort of thing in the coming years, as hexagons rise to ever greater prominence in the collective human experience, perhaps leading at some point to the formation of an explicitly "hexagonalist" movement in art and design.

As the harvest season draws nigh on to a close here in the northern climes, I thought it might be a good time to take a look at the year's HEXAGONAL CROP CIRCLES.

Now, I try not to get overly woo-woo here, so let me preface my remarks by assuring you, the gentle reader, that, pending the advent of more compelling evidence, I continue to maintain a strict agnosticism towards the phenomenon of crop formations. But I am comfortable making at least the following two assertions on the subject:

1) There are at least some crop formations from the past thirty years that simply were not made covertly, at night, by a small group of people with boards and crap. Not all of them, maybe not even most of them, but definitely some of them.

2) There is a spectrum of possible explanations for these formations between and beyond the false dichotomy of "pranksters running around fields at night with boards and wires" versus "zOMG ALIENS!" usually put forward by mainstream media and other consensus-reality-builders in our society.

Indeed, as an aside, I would like to point out that I find it highly unfortunate that both crop circles and the more well-documented UFO phenomenon have come to be associated, for little reason, with theories of extraterrestrial visitation. In the case of UFOs, the subjects are conflated so completely that you often hear people asking if one "believes in" UFOs, when they actually mean "Do you believe aliens are visiting Earth in nuts-and-bolts spacecraft from other star systems?" Which are obviously two completely different questions. (And how one could possibly "not believe" there are flying objects that are not, in fact, "identified," is itself utterly beyond my comprehension.) In the case of UFOs, at least there are actual apparent flying vehicles involved, so I can understand the conceptual leap. I have never understood at all how extraterrestrial speculation came into the crop circle issue, except by association with UFO culture, and a general lack of imagination as to the different ways intelligence may manifest itself in this universe. I mean, it could be extraterrestrials. It would be interesting if it were. But this is certainly not by default assumption, nor is it even remotely high on my list of plausible explanations.

Word on the street is that today's Google Doodle features an interactive truncated icosahedron in recognition of the 25th(.) anniversary of the discovery of the buckminsterfullerene (and, by extension, of fullerenes, nanotubes, &c. in general).

I normally don't actually pay attention to doodles—I find them rather played out at this point, and with the rise of integrated browser search fields, who even goes to Google's home page anymore. But it is worth noting that—according to sources who follow these things more closely than I do—this is ostensibly only Google's second "interactive" doodle, after this past spring's Pac-Man episode. THIS IS NOT AS INTERESTING AS PAC-MAN, but it's presumably indicative of the moral and intellectual gravity they assign to the discovery of fullerenes that they would consider the event worthy of such rarified and exotic treatment. The spinning of the fullerene here is not merely meant to amuse us for half a second, but rather to instill in us an understanding that the event being commemorated is, on balance, probably more important than, you know, the Australian federal election, or Chinese Valentine's Day, or what have you.

While reviewing the hexagonal news this morn, I ran across an article describing some sort of Hexagon Project out of Scranton, Pennsylvania. Looking into it further, I expected to find that—like many hexagon-identified things these days—the "hexagon" was a purely metaphorical branding feature. To my surprise though, it not only involves literal hexagons, but the hexagonal geometry seems to have been deliberately and consciously chosen for very hexagonally-aware reasons. The project's page explains, under the heading "Why a hexagon?":

"The hexagon is a composition of complex relationships, interdependent lines, like bonds of human connection, strengthened in multiples into an infinite network of connections. It maintains its own presence as a shape, symbol of light and life, yet, structurally, destined to be part of a whole—a splendid architectural element, infinitely expandable."

[Edit 2013-10: The existence of this post has bothered me since I first published it three years ago. Basically, this was intended as the first installment of what was to have been a sprawling series of posts attempting to reconcile what I will loosely term "Eastern monism" with the semantically precise, mathematical ontology I have been trying to develop in light of hexagonal principles—and in a real, meaningful way, not just in some fake-ass pop quantum mysticism sense. I won't recapitulate my whole line of thought here, but suffice to say I had and continue to have more substantive and I think relevant ideas on these matters than those expounded in this limited post, which by itself I think ends up conveying either that I misunderstand or misinterpret people like Watts, or that I am not particularly interested in the finer implications of what they tried to communicate. And I would like to think neither is true. Watts was a huge intellectual influence on me in my youth, and it was writings such as his that actually turned me from my earlier phase of hexagonal thinking as a teenager—only to return to it with more clarity of thought over a decade later. And having later become of aware of his passing references to hexagons, I was very eager to draw out these ideas and sort of riff on them with the greater perspective gained from the ensuing years. But I left the task unfinished, and, again, I think this post, by itself, is sort of weird. I gave serious thought to deleting it or leaving it unpublished in this latest iteration of the site, but ultimately decided to leave it here, with this disclaimer. I intend to revisit these issues at some point, hopefully in greater detail, at which point I will perhaps modify or edit this post further.]

"Where is it writ large that talking monkeys should be able to model the cosmos?" – Terence McKenna

I remember reading a short but eloquent endorsement of hexagons by Alan Watts once, probably seven or eight years ago. Since I wasn't particularly interested in hexagons at the time, I quickly forgot about it, and where exactly I had read it. However, times have changed and my hexagonal thinking has evolved, and with the advent of Google Books and all, I recently decided to look into the matter further. After an exhaustive search, I found two mentions of hexagons in Watts' books, which I shall now share with you, and duly expound upon.

I am not entirely sure either of these is in fact the passage I remember reading, but if it were either of them it would be this first one, which can be found in the essay What on Earth Are We Doing?, from the collection Cloud-Hidden, Whereabouts Unknown:

The last components necessary to make this site passably acceptable to me were completed last evening. Thus, as promised, on time and under budget, I hereby declare this site to exist. Several important facts to keep in mind:

First and foremost, the site is of course not really done. Also it still sort of sucks. But it is functional enough that it stands on its own, does not have any overtly malfunctioning content, and generally seems to work pretty well, so rather than delaying any longer I am just going to put it out there. But I expect to be adding many useful and pertinent features and content items in the coming weeks and months. So check back often or something.

(For a reasonably convincing argument on why the letter τ (tau) in particular should be adopted for this value, please read Mr. Hartl's manifesto.)

The fundamental point here is that, in trigonometry and all other manner of angle-measuring endeavors, what we care about is the radius of a circle, not its diameter. The one follows from the other to be sure, but at the end of the day the diameter is more usefully considered twice the radius than the radius is half the diameter. A circle is a circumference around a center—it is the measure of this distance between center and circumference that is elemental to the idea of a circle, not the rather incidental fact that its full width is twice that same distance.

I seem to have taken an unintentional hiatus from putting my ass into this project, due largely to the fact that it has of late been too hot to move, think, or do anything at all really. Come hell or high water though, I will "launch" this site by the first of Sextilis, being actually the fifteenth (13th;) anniversary of the launch of my first Hexnet site, back in 1995. That is probably an unnecessarily dramatic theme to tie in with this whole procedure, but it seems silly to allow such an auspicious date to pass unobserved. Also, I really need to get on with this. I mean, I have numerous medium-term goals to tweak the layout—I want to clean up and expand the links section, among other things—but most of that can wait. I am really just trying to wrap up a bit of library content, actually. Which at this point seems fairly silly, since I long ago gave up on my original idea of having some sort of centralized "Hexagonal Manifesto" or what have you, in favor of a more modular if somewhat disjointed collection of one-off content items and such. Anyway, since 1 August falls lamentably on a Sunday this year, which is a terrible day to do anything PR-wise (as is the entire month of August, really), I expect to do some sort of formal what have you by next week some time. Or you know whenever.

Anyone following the hexagonal news of late has no doubt noticed a flurry of stories related to the upcoming release of Civilization V, and in particular its new hexagonal tile system. In this blog's opinion, the adoption of hexagonal tiles by the Civ franchise is a long overdue development, and frankly one that should've been incorporated into at least the last two Civ releases. Indeed, many DOS-based strategy games have used hexagonal tiles going back to the late '80s, and one has to wonder why Civilization ever used square tiles. (I remember playing the original Civilization in the early '90s, well before the advent of my own hexagonal illumination, and thinking that, in fact, it would be better with a hexagonal grid. I can only imagine part of the media excitement here—aside from just the general awesomeness of hexagons—is due to the fact that many, many other people have had the exact same thought over the past twenty years.)

I have been reading about the PAH world hypothesis, and have come to see it as an intriguing indicator of the potentially hexagonal origins of life on earth.

Essentially, it is conjectured that, since polycyclic aromatic hydrocarbons are among the most common spaceborne molecules in the known universe, they would have likely been a constituent in the primordial seas of Earth, where they could have provided some sort of scaffolding or template on which early biological polymers such as RNA could assemble, thus solving a frequently-raised objection to the RNA world hypothesis that RNA is too fragile and transient to survive long outside of an extant cell or similar protective environment. By providing a structural backbone on which reasonably complex RNA strands and such could self-assemble, the PAH world would have given early pre-cellular life a fighting chance of finding its way into protective lipid bubbles, weird mineral formations, or what have you, where given enough replicative iterations it presumably developed into proper cellular life as we know it.

It looks like the hexagonal zeitgeist is finally floating to the surface of our collective media environment. Two stories in particular off of our hexagonal news feed caught my eye this morn:

Geometry lessons
". . . Just as the 1950s will always be remembered as the decade of atomic-inspired motifs, so may the early decades of the 21st century be remembered, by those of pop cultural bent, as belonging to the hexagon."

The Hexagon Harley 1.4301
"Six is the magic number on this handcrafted custom Harley Davidson by Horst Dzhangmen. Everything but the engine, a HD Shovelhead 1340 cm3 and the frame, has been cut and built into perfectly shaped hexagons and this exceptional work took its creator three years of hard work to achieve this perfection."

This site is not yet completed, and at least one core content piece is still in progress, but after some months of halfassery, I am prepared at this time to move into Phase I of my three-phase site launch program. This site is live as of May 30, 2010 (May 26, 11B6), and hopefully shall remain so. FTVW.

Again, I cannot overemphasize how not-ready-for-formal-launch this site is. Structurally, several site elements still need some work, and as I said some content is still being developed, including my short mathematical summary of hexagons, which, though somewhat trivial, arguably constitutes a somewhat significant piece of content on a site nominally about hexagons.

Hopefully we will be ready to proceed to Phase II within the next week. Thank you.

I have been completely unable to find any sort of followup to this since 2003. The whole idea may have been discarded by now, I'm not sure, but considering that Plato speculated—for no particular reason modern science is aware of—that the universe was shaped like a dodecahedron nigh doz. 1,500 years ago, this is obviously a significant conjecture to put forward.

While the other four platonic solids have traditionally been associated with the four classical elements, the dodecahedron has been identified with the universe, or the aether. Did the ancients have some insight into the cosmological structure of our universe that we are only rediscovering now? Maybe! Or maybe the whole idea is crap. Who knows! But the hexagonal implications are obvious—twelve being a multiple of six, the twelve faces of the dodecahedron forming six symmetrical pairs, et cetera.

The following video came to my attention recently. It presents, in my view, a perfect example of the sort of world-class hexagonal education we once provided our children in that bastion of cultural exceptionalism known as the 1980s, and which seems sadly lacking from today's undoubtedly clever yet somehow less challenging children's programming:

Interesting article from Science's popular science outfit about the Saturnine hexagon. While it is certainly an intriguing development, and a step forward from the well-known "spinning bucket" experiments, they should get back to us when they've made a hexagon 15,000 miles wide that lasts for 30 years. (Which, for practical purposes, one wouldn't expect any time soon.) Fluid dynamics can be vastly different at different scales. I would also like to know what the viscosity is of this "water" they speak of relative to the atmosphere of Saturn.

The following is a short writeup I did for an event on Facebook celebrating Dozenal Pi Day on March 18, 11B6 (decimal March 20, 2010). As indicated in the text, it includes some borrowed material from the Wikipedia article on duodecimalism.

I am not sure when exactly this "Pi Day" thing became so fashionable, but for me it has come to symbolize everything that is wrong and unwholesome about modern society's relationship with numbers. You see, of late I find myself more and more perturbed by the extent to which people seem to take the primacy of decimal notation for granted in conceptualizing and comparing numbers. In the case of pi, there is NOTHING particularly universal or ontologically significant in the sequence of digits 3.14159265... except that they represent a certain way of parsing the value of pi by successive fractions of ten. THIS IS MAY BE INTERESTING IF YOU ARE REALLY INTO THE NUMBER TEN FOR SOME REASON. Otherwise it—like the entire decimal radix system—is fairly useless.